An astronaut at some time in the future has travelled to a distant world around

Question

An astronaut at some time in the future has travelled to a distant world around the double star Luyten, which the folks back home measure as 9 light years away from earth. While on that planet, he purchased a pair of Kurinji plants, that bloom every 12 years, a gift for his wife who has been home raising the kids and running a major corporation. The plants had just bloomed when he bought them, so it will be 12 years until they bloom again. He keeps one at the window near the fron of the ship and one near the back. He arrives home rather pleased and somewhat surprised that his plants are just in bloom. His wife has calculated that, given the spacecraft he is using(which has a rest mass of 5*10^8 kg), it will be 15 years for him to get to her(at least as she sees it, and she has a PhD in math from Princeton. To the astronaut's wife the plants are flowering just as he is getting home because A: Time for the plants ran slow B:Time for the plants ran fast The answer is A but I am looking for a short explanation as to why it is A.

Explanation / Answer

To the astronaut, the plants are flowering just as he is getting home because the distance home was a shorter disatnce than expected.

To the astronaut's wife, the plants are flowering just as he is getting home because -

(A) Time for the plants ran slow

The astronaut's ship travelling at a speed which will be given as :

using a formula, we have

v = d / t = (9 c-yr) / (15 years)

v = 0.6 c

where, c = speed of light = 3 x 108 m/s

then, we get

v = 1.8 x 108 m/s

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